# Thread: equation of a reflected line

1. ## equation of a reflected line

We know that suppose a light with the equation $y=2x$ hit a mirror (let it be x-axis [ $y=0$]), a resultant ray will have an equation of $y=-2x$.

Now let say if I have a light ray of equation $y=10x+5$ and the mirror equation $y=3x+3$. Then what will be the resultant equation of the reflected ray?

Attached is the diagram. Not drawn to scale

2. y=(-43/26)*x+(152/91)

3. Originally Posted by SebEndo
y=(-43/26)*x+(152/91)
Hi Sebendo,

May I know how you approach the question in order to get the answer?

4. oh, yes sorry about that, I though you just wanted the answer.

There are different ways of solving this: you could use parametric equations of the line you have and then tunr it into matrices but what I did is this:

First find the intersection point: (15/17 , -2/7) (let me know if you have trouble finding this.

Next, because it is a reflected light beam we know the relation between angles: the angle of incidence will be equal to the angle of reflection. let the angle of the mirror line with the horizontal be A and the angle of the beam with the horizontal be B. Draw this out and label the angles and you will find that the angle between the tow line (C) is C=B-A i)

Then, if you draw a horizontal line from the point of intersection you can see that the angle of the reflected beam (D) is: D=C-A ii)

use i) and ii) to isolate D: D=B-2A iii)

lastly we know that the angle of a line with the horizontal is the arctan of its slope. therfore we state that D=arctan(m) ,m being the slope of the reflected light beam. Using iii) we get the folowing equation:
=> arctan(m)=arctan(10)-2*arctan(3)
=> m=tan(arctan(10)-2*arctan(3)) = -43/26

now you have a point and the slope of the light beam. plug this in y=mx+b, solve for b and you have you equation.

I hope this is not too confusing, its only my second post...

good luck

5. Originally Posted by SebEndo
First find the intersection point: (15/17 , -2/7) (let me know if you have trouble finding this.

Next, because it is a reflected light beam we know the relation between angles: the angle of incidence will be equal to the angle of reflection. let the angle of the mirror line with the horizontal be A and the angle of the beam with the horizontal be B. Draw this out and label the angles and you will find that the angle between the tow line (C) is C=B-A i)

Then, if you draw a horizontal line from the point of intersection you can see that the angle of the reflected beam (D) is: D=C-A ii)

use i) and ii) to isolate D: D=B-2A iii)

lastly we know that the angle of a line with the horizontal is the arctan of its slope. therfore we state that D=arctan(m) ,m being the slope of the reflected light beam. Using iii) we get the folowing equation:
=> arctan(m)=arctan(10)-2*arctan(3)
=> m=tan(arctan(10)-2*arctan(3)) = -43/26
Hi Thanks for your clarification again.

I got $(-\frac{2}{7}, 2\frac{1}{7})$ as my interception. Maybe you can explain how to get it as well. Thanks

I am quite confused with your definition of the angles. Attached is my understanding of your explanation. Please verify again Tks

6. ## correction

I am sorry you are right, I did it too fast and inverted the 2 and 7 in x (it was pretty late too ) but the resoning work you can redo the work with the right point.

angle clarification: angles A B and D are angles with the horizontal counter clockwise (or in your case, the smallest ones). and angle C is the smallest angle between the mirror and the light beam.

7. Originally Posted by SebEndo

angle clarification: angles A B and D are angles with the horizontal counter clockwise (or in your case, the smallest ones). and angle C is the smallest angle between the mirror and the light beam.
Can u check with the new attached drawing again? Still not sure where is the D

8. Originally Posted by noob mathematician
Can u check with the new attached drawing again? Still not sure where is the D
yep that right, you got the picture. You got the angle D properly drawn also.

9. Here, maybe this will help: